Phys 222
Exploring Electric Potentials & Electric Fields
This is a group write-up. One group should include 3 or 4 students.
Introduction & Objectives:
The focus in this section is in the area of electrostatics and uses the concept of electric
potential to explain our experience of many electrical phenomena such as:
- how insulators become conductors
- getting shocked
- lightning
- effects of pointed vs. round objects, etc.
Applications of electric potentials in electrostatics include:
-
cathode ray tubes in oscilloscopes
electric lenses
fluorescent lights
television tubes
electrostatic shielding, etc.
We will use the EM Field simulation software to visually explore the nature of electric potentials
and electric fields. In this exploration we will determine the electric field from an understanding
of the electric potential. But we could just as well go the other direction. One dimensionally
speaking the electric field is the derivative of the electric potential with respect to position. The
fundamental theorem of calculus tells us that by using integration we can go from the electric
field back to the electric potential.
Ex = -dV/dx
V=-
 E  dx
We'll use the EM Field program to determine shapes, magnitudes and superposition of
equipotentials Then we'll study how equipotentials can give valuable information on the
magnitude and direction of the electric field.
Electric Potential & Equipotentials:
Electric potential is a scalar quantity. The value of the electric potential at a position near a
charged particle or object gives a relative value for the electrical potential energy per unit charge
at that position. The units for electric potential are then "energy/charge" or in the MKS system
Joule/Coulomb = Volt. Depending upon the shape of the charged object there will exist points
near it that have the same electric potential. The locus of these points of equipotential form a
curve or surface surrounding the charged object that looks like a halo. The simulation program is
two dimensional and draws equipotentials as you use the mouse to click at points in the region
surrounding the charges. The equipotential curves describe the electrical potential landscape
near the charges in the same way that a topological map describes the gravitational potential
landscape.
Part I: Equipotentials of a Point of Charge
Procedure:
A.
Use EM Field to set up a positive point charge in the center of the screen. Include the
grid.
Under the "Sources" menu select "3D Point Charges"
Under the "Display" menu select "Show grid"
Click and drag any positive charge to the center of the screen.
B.
Click on grid points along a horizontal line going through the charge to display the shape
and value of the equipotentials in the region of the charge.
Under the "Field & Potential" menu select "Equipotential with Number".
C.
Print a copy of your picture.
Exercises & Questions:
1.
Dependence of Electric Potential on Distance from the Charge:
Use your printout of the screen to create a graph of electric potential vs. distance from
the charge. The units of potential are volts. The grid is in centimeters. Create a fit that
describes how electric potential depends on distance. Describe in writing the nature of
the relationship between electric potential and distance from the charge.
2.
Dependence of Equipotentials on Magnitude & Sign of Charge:
Predict in writing what changes in the equipotential lines might occur if the magnitude of
the charge is changed. What if the sign of the charge changes? Test your
predictions by using EM field to create an equipotential map of the opposite
charge that you just printed above. Print the result and create graphs of V vs.
x for the negative charge. Write about the results. Refer to printed copies of your
results.
Part II: Superposition of
Equipotentials
Procedure:
A.
Use the data gathered in the previous exercise to calculate the magnitude of the electric
potentials at grid points along a horizontal axis running through the two charges if they
were positioned as a dipole with four centimeters between them. You’ll need to make a
careful sketch on a blank piece of paper. You can make the calculations by overlaying
the equipotential maps made in the previous exercise of a single positive and negative
charge of equal magnitudes (separated by 4 cm) and summing their potentials at key
points on axis and off axis. Show how you made these calculations. Plot the values of
equipotentials that you made on a diagram. Show your results to your instructor
before you proceed.
B.
Use EM Field to set up the dipole situation described above of two equal and opposite
charges on the screen with 4 grid points between them. The charges should have the
same magnitude as those you created in part A.
Click on grid points along an horizontal axis running through the two charges to display
the shape and value of the equipotentials in the region of the charges.
C.
G.
Print a copy of your data.
Exercises & Questions:
1.
Compare Predicted Shape for Equipotentials with EM Field:
How did your initial prediction of shape of the equipotentials for the dipole compare with
EM Field's calculations? How close did you come to predicting the shape of the
equipotentials?
2.
Describe the meaning of the equipotential map that you made. Where are the
equipotentials close together vs. far apart and what does that mean about the behaviour
of a charged particle in those regions?
2.
Equipotentials for Equal Charges of the Same Sign:
a. Predict the shape of the equipotentials and values of electric potential along the
horizontal for a pair of charges of like sign and magnitude.
b. Test your prediction. Discuss the result. Compare this result to the one of
opposite charges. Include your prediction sketch and the printed sketch.
c. Describe the meaning of this equipotential map. Where are the equipotentials close
together and where are they far apart and what does this mean about the behaviour
of a particle in those regions?
Electric Fields & Electric Field Lines
Definition:
As described in the introduction an electric field can be determined from the electric potential by
taking the derivative of the potential with respect to position. More correctly the electric field is
defined to be the gradient of the electric potential which for rectangular symmetries would be
calculated:
Ex = -dV/dx
Ey= -dV/dy
Ez = -dV/dz
OR
for spherical symmetries.
Er = -dV/dr
The electric field is a vector which for a particular position describes the magnitude and direction
of the electric force per unit charge at that position. The units of the electric field are
consequently "Force/Charge" or in the MKS system "Newtons/Coulomb". Physically speaking
the electric field gives the magnitude and direction of maximum change in the electric potential.
For example the electric field will not be directed parallel to an equipotential line because by
definition the electric potential is not changing along this line.
We will use EM Field to show that the electric field of a charged object can be easily determined if
the equipotential lines are already known for the object. In exploring the connection between
electric fields and electric potentials it can be helpful the keep in mind the analogous relationship
that exists between gravitational potential and gravitational force.
Electric field lines show the direction of the electric force on a positive "test" charge in the region
of the charged object. The magnitude of the electric field is changing on these lines. The
magnitude of the electric field is a constant along equipotentials. Michael Faraday, the scientist
who invented electric field lines thought of them as "rubber bands". If you increased the tension
in the field lines you could sense the attraction or repulsion between the charges in the region.
Part III: Electric Fields & Electric Field Lines - Point Charge
A.
Use EM Field to set up a magnitude 5 or larger positive point charge in the center of the
screen. Include the grid.
B.
Click on grid points along a horizontal line going through the charge to display the shape
and value of the equipotentials in the region of the charge.
(Under the "Field & Potential" menu select "Equipotential with Number".)
C.
Click on grid points along a horizontal line going through the charge to display the electric
field vectors in the region of the charge.
(Under the "Field & Potential" menu select "Field Vector".)
D.
Click on grid points around the charge to display the electric field lines in the region of
the charge.
(Under the "Field & Potential" menu select "Field lines".)
E.
Print a copy of your picture.
Exercises & Questions:
1.
Relationship between the electric field and the electric potential:
The magnitude of the electric field can be approximated by calculating how the electric
potential is changing from one position to another. Using the printout of the potential
data you've collected, calculate for every grid point (1 cm or .01m increments) the
electric field:
E = - V/x = - (V2 - V1)/.01m
Create a graph of the values and a curve fit of the data. Compare your result to what
you know about the how the electric field of a point charge depends on distance from the
charge.
2.
Electric field lines and equipotential lines:
What relationship can you see between the equipotential lines and electric field lines?
What do the field lines and equipotentials look like for regions of weak electric field?
What do the field lines and equipotentials look like for regions of strong electric fields?
Discuss in terms of the data you've collected and their mathematical relationship.